A004331 Binomial coefficient C(4n,n-1).

1, 8, 66, 560, 4845, 42504, 376740, 3365856, 30260340, 273438880, 2481256778, 22595200368, 206379406870, 1889912732400, 17345898649800, 159518999862720, 1469568786235308, 13559593014190944, 125288932441604200

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A002293, A005810, A052203, A224274, A257633, A262977.

Cf. A001791, A004319, A004343, A004356, A004369, A004382.

Programs

Maple

#A004331
seq(binomial(4*n - 1,n), n = 0..20);

Mathematica

a[n_] := Binomial[4*n, n - 1]; Array[a, 19] (* Amiram Eldar, May 09 2020 *)

PARI

vector(30, n, binomial(4*n, n-1)) \\ Altug Alkan, Nov 05 2015

Formula

G.f.: (g^2-g)/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011

With an offset of 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A257633 (k = 2) and A224274 (k = 3). - Peter Bala, Nov 04 2015

D-finite with recurrence 3*(n-1)*(3*n-1)*(3*n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 19 2025